I have a bivariate objective function that is strictly concave but too complex to derive a closed-form solution from the first-order condition. As a bypath, I want to approximate the objective function only up to the second-order and derive its solution.
Is it even plausible? Is there any mathematical way to calculate the boundary of the error?
I would like to specify my maximization problem as follows.
I have a strictly concave function $W(x,y)$. I know that $W(x,y)$ is maximized at, for example, $(X,Y)$ where $X \neq Y$. I want to compare the maximum with one on which I add a constraint that $x=y$. Then $W(z,z)$ would be strictly inferior to $W(X,Y)$ for any $z$.
Now I turn $max$ $W(z,z)$ to $max$ $W(z,z)-W(X,Y)$, as the latter term is constant. Now I approximate $W(z,z)$ to $W(X,Y)$. Then the approximation would be $W_{1}(z-X)+W_{2}(z-Y)+0.5W_{11}(z-X)^{2}+0.5W_{22}(z-Y)^{2}+W_{12}(z-X)(z-Y)...$.
I disregard any higher order than the second. And maximize only $W_{1}(z-X)+W_{2}(z-Y)+0.5W_{11}(z-X)^{2}+0.5W_{22}(z-Y)^{2}+W_{12}(z-X)(z-Y)$ with respect to $z$. However, as we defined $(X,Y)$ as the argmax, $W_{1}$ and $W_{2}$ vanish. Therefore, we can consider only the second order.
Thanks